Optimal. Leaf size=185 \[ -\frac {a^2 c^3 x}{d^3}-\frac {4 a b c^3 (c+d x)^{3/2}}{3 d^4}+\frac {c^2 \left (3 a^2-b^2 c\right ) (c+d x)^2}{2 d^4}+\frac {12 a b c^2 (c+d x)^{5/2}}{5 d^4}-\frac {c \left (a^2-b^2 c\right ) (c+d x)^3}{d^4}-\frac {12 a b c (c+d x)^{7/2}}{7 d^4}+\frac {\left (a^2-3 b^2 c\right ) (c+d x)^4}{4 d^4}+\frac {4 a b (c+d x)^{9/2}}{9 d^4}+\frac {b^2 (c+d x)^5}{5 d^4} \]
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Rubi [A]
time = 0.18, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {378, 1412, 786}
\begin {gather*} \frac {c^2 \left (3 a^2-b^2 c\right ) (c+d x)^2}{2 d^4}+\frac {\left (a^2-3 b^2 c\right ) (c+d x)^4}{4 d^4}-\frac {c \left (a^2-b^2 c\right ) (c+d x)^3}{d^4}-\frac {a^2 c^3 x}{d^3}-\frac {4 a b c^3 (c+d x)^{3/2}}{3 d^4}+\frac {12 a b c^2 (c+d x)^{5/2}}{5 d^4}+\frac {4 a b (c+d x)^{9/2}}{9 d^4}-\frac {12 a b c (c+d x)^{7/2}}{7 d^4}+\frac {b^2 (c+d x)^5}{5 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int x^3 \left (a+b \sqrt {c+d x}\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sqrt {x}\right )^2 (-c+x)^3 \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int x (a+b x)^2 \left (-c+x^2\right )^3 \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int \left (-a^2 c^3 x-2 a b c^3 x^2-c^2 \left (-3 a^2+b^2 c\right ) x^3+6 a b c^2 x^4+3 c \left (-a^2+b^2 c\right ) x^5-6 a b c x^6+\left (a^2-3 b^2 c\right ) x^7+2 a b x^8+b^2 x^9\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=-\frac {a^2 c^3 x}{d^3}-\frac {4 a b c^3 (c+d x)^{3/2}}{3 d^4}+\frac {c^2 \left (3 a^2-b^2 c\right ) (c+d x)^2}{2 d^4}+\frac {12 a b c^2 (c+d x)^{5/2}}{5 d^4}-\frac {c \left (a^2-b^2 c\right ) (c+d x)^3}{d^4}-\frac {12 a b c (c+d x)^{7/2}}{7 d^4}+\frac {\left (a^2-3 b^2 c\right ) (c+d x)^4}{4 d^4}+\frac {4 a b (c+d x)^{9/2}}{9 d^4}+\frac {b^2 (c+d x)^5}{5 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 105, normalized size = 0.57 \begin {gather*} -\frac {16 a b \sqrt {c+d x} \left (16 c^4-8 c^3 d x+6 c^2 d^2 x^2-5 c d^3 x^3-35 d^4 x^4\right )+315 a^2 \left (c^4-d^4 x^4\right )+63 b^2 \left (c^5-5 c d^4 x^4-4 d^5 x^5\right )}{1260 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 78, normalized size = 0.42
method | result | size |
default | \(b^{2} \left (\frac {1}{5} d \,x^{5}+\frac {1}{4} c \,x^{4}\right )+\frac {4 a b \left (\frac {\left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {3 c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {3 c^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {c^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}\right )}{d^{4}}+\frac {a^{2} x^{4}}{4}\) | \(78\) |
trager | \(\frac {\left (4 b^{2} d x +5 b^{2} c +5 a^{2}\right ) x^{4}}{20}-\frac {4 a b \left (-35 d^{4} x^{4}-5 c \,d^{3} x^{3}+6 c^{2} x^{2} d^{2}-8 c^{3} d x +16 c^{4}\right ) \sqrt {d x +c}}{315 d^{4}}\) | \(81\) |
derivativedivides | \(\frac {\frac {b^{2} \left (d x +c \right )^{5}}{5}+\frac {4 a b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {\left (-3 b^{2} c +a^{2}\right ) \left (d x +c \right )^{4}}{4}-\frac {12 c a b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (3 b^{2} c^{2}-3 a^{2} c \right ) \left (d x +c \right )^{3}}{3}+\frac {12 c^{2} a b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {\left (-b^{2} c^{3}+3 a^{2} c^{2}\right ) \left (d x +c \right )^{2}}{2}-\frac {4 c^{3} a b \left (d x +c \right )^{\frac {3}{2}}}{3}-c^{3} a^{2} \left (d x +c \right )}{d^{4}}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 151, normalized size = 0.82 \begin {gather*} \frac {252 \, {\left (d x + c\right )}^{5} b^{2} + 560 \, {\left (d x + c\right )}^{\frac {9}{2}} a b - 2160 \, {\left (d x + c\right )}^{\frac {7}{2}} a b c + 3024 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c^{2} - 1680 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{3} - 1260 \, {\left (d x + c\right )} a^{2} c^{3} - 315 \, {\left (3 \, b^{2} c - a^{2}\right )} {\left (d x + c\right )}^{4} + 1260 \, {\left (b^{2} c^{2} - a^{2} c\right )} {\left (d x + c\right )}^{3} - 630 \, {\left (b^{2} c^{3} - 3 \, a^{2} c^{2}\right )} {\left (d x + c\right )}^{2}}{1260 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 94, normalized size = 0.51 \begin {gather*} \frac {252 \, b^{2} d^{5} x^{5} + 315 \, {\left (b^{2} c + a^{2}\right )} d^{4} x^{4} + 16 \, {\left (35 \, a b d^{4} x^{4} + 5 \, a b c d^{3} x^{3} - 6 \, a b c^{2} d^{2} x^{2} + 8 \, a b c^{3} d x - 16 \, a b c^{4}\right )} \sqrt {d x + c}}{1260 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.73, size = 139, normalized size = 0.75 \begin {gather*} \begin {cases} \frac {\frac {a^{2} d x^{4}}{4} + \frac {4 a b \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{3}} + \frac {2 b^{2} \left (- \frac {c^{3} \left (c + d x\right )^{2}}{4} + \frac {c^{2} \left (c + d x\right )^{3}}{2} - \frac {3 c \left (c + d x\right )^{4}}{8} + \frac {\left (c + d x\right )^{5}}{10}\right )}{d^{3}}}{d} & \text {for}\: d \neq 0 \\\frac {x^{4} \left (a + b \sqrt {c}\right )^{2}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.63, size = 151, normalized size = 0.82 \begin {gather*} \frac {252 \, b^{2} d^{2} x^{5} + 315 \, b^{2} c d x^{4} + 315 \, a^{2} d x^{4} + \frac {144 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a b c}{d^{3}} + \frac {16 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b}{d^{3}}}{1260 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.37, size = 167, normalized size = 0.90 \begin {gather*} \frac {b^2\,{\left (c+d\,x\right )}^5}{5\,d^4}-\frac {\left (6\,b^2\,c-2\,a^2\right )\,{\left (c+d\,x\right )}^4}{8\,d^4}+\frac {\left (6\,a^2\,c^2-2\,b^2\,c^3\right )\,{\left (c+d\,x\right )}^2}{4\,d^4}-\frac {a^2\,c^3\,x}{d^3}+\frac {4\,a\,b\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}+\frac {c\,\left (b^2\,c-a^2\right )\,{\left (c+d\,x\right )}^3}{d^4}-\frac {4\,a\,b\,c^3\,{\left (c+d\,x\right )}^{3/2}}{3\,d^4}+\frac {12\,a\,b\,c^2\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}-\frac {12\,a\,b\,c\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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